Properties of Cubes of Numbers
Properties of Cubes of Numbers
The cube of natural numbers have the following interesting properties:
Property 1: Cubes of all even natural numbers are even.
Property 2: Cubes of all odd natural numbers are odd.
Property 3: The sum of the cubes of first n natural numbers is equal to the square of their sum. That is,
Property 4: Cubes of the numbers ending in digits 1, 4, 5, 6, and 9 are the numbers ending in the same digit. Cubes of numbers ending in digit 2 end in digit 8 and the cube of numbers ending in digit 8 ends in digit 2. The cubes of the numbers ending in digits 3 and 7 ends in digits 7 and 3 respectively.
Explanation: The cubes of the first 10 natural numbers are given in the following table.
| Number x | Cube |
Number x | Cube |
| 1 | 1 | 11 | 1331 |
| 2 | 8 | 12 | 1728 |
| 3 | 27 | 13 | 2197 |
| 4 | 64 | 14 | 2744 |
| 5 | 125 | 15 | 3375 |
| 6 | 216 | 16 | 4096 |
| 7 | 343 | 17 | 4913 |
| 8 | 512 | 18 | 5832 |
| 9 | 729 | 19 | 6059 |
| 10 | 1000 | 20 | 8000 |
We shall learn about the cubes of negative integers.
We have,
is the cube of itself.
Similarly,
is the cube of -2.
is the cube of -3 and so on.
In general, if m is a positive integer, then
Thus, for any positive integer is the cube of -m.
Cube of a Rational Number
Let be a rational number (m, n are non-zero integers such that
) other than an integer, then the cube of a is defined as
or,
| |||
| Right Option : B | |||
| View Explanation | |||
The square of a natural number subtracted from its cube is 48. The number is ___________________. | |||
| Right Option : C | |||
| View Explanation | |||
The cube of a number is 8 times the cube of another number. If the sum of the cubes of numbers is 243, the difference of the numbers is : | |||
| Right Option : A | |||
| View Explanation | |||
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